Lionel Schwartz

Abstract.
This paper proves a particular case of a conjecture of N. Kuhn. This
conjecture is as follows. Consider the Gabriel-Krull filtration of the
category U of unstable modules.
Let U_n, n>=0, be the n-th step of
this filtration. The category U is the smallest thick sub-category
that contains all sub-categories U_n and is stable under colimit
[L. Schwartz, Unstable modules over the Steenrod algebra and
Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics
Series (1994)]. The category U_0 is the one of locally finite modules,
i.e. the modules that are direct limit of finite modules. The
conjecture is as follows, let X be a space then :
* either H^*X is locally finite,
* or H^*X does not belong to U_n, for all n.
As an example the cohomology of a finite space, or of the loop space
of a finite space are always locally finite. On the other side the
cohomology of the classifying space of a finite group whose order is
divisible by 2 does belong to any sub-category U_n. One proves this
conjecture, modulo the additional hypothesis that all quotients of the
nilpotent filtration are finitely generated. This condition is used
when applying N. Kuhn's reduction of the problem. It is necessary to
do it to be allowed to apply Lannes' theorem on the cohomology of
mapping spaces.[N. Kuhn, On topologically realizing modules over the
Steenrod algebra, Ann. of Math. 141 (1995) 321-347].